
where \(\alpha = 3\)  cells [1]}, \(\beta = 3.5\) , \(\eta = 4\)  cells, and \(N^\text{d}_\text{I}\)  is the number of \(I^\text{d}\)  cells. Using the same notation, the corresponding CA rule for \(M\)  differentiation at the randomly selected grid position \(\textbf {z}\)  is:
\(&\underbrace{\sum _{i=1}^{N^\text{d}_\text{X}} \textbf {1}_{\Omega _\text{long}^\textbf {z}} (\textbf {X}_i^\text{d}) + \sum _{i=1}^{N^\text{d}_\text{I}} \textbf {1}_{\Omega _\text{long}^\textbf {z}} (\textbf {I}_i^\text{d}) > \alpha + \beta \sum _{i=1}^{N_\text{M}} \textbf {1}_{\Omega _\text{long}^\textbf {z}} (\textbf {M}_i)}_\text{long-range signals (same as Eqn. (\ref {eq:rule1}))} \text{~~~and} \\&\underbrace{\sum _{i=1}^{N^\text{d}_\text{X}} \textbf {1}_{\Omega _\text{loc}^\textbf {z}} (\textbf {X}_i^\text{d}) \le \gamma \sum _{i=1}^{N_\text{M}} \textbf {1}_{\Omega _\text{loc}^\textbf {z}} (\textbf {M}_i) \text{~and~} \sum _{i=1}^{N^\text{d}_\text{I}} \textbf {1}_{\Omega _\text{loc}^\textbf {z}} (\textbf {I}_i^\text{d})\le \kappa }_\text{local signals: competition} \text{~~} \Longrightarrow \text{~~$M$ differentiates at $\textbf {z}$}, \nonumber \) 
